Chapter 2

\(\mathrm{A}=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]\), then \(A^{n}\) is (a) \(\left[\begin{array}{cc}1+2 n & -4 n \\ n & 1-2 n\end{array}\right]\) (b) \(\left[\begin{array}{cc}3^{n} & (-4)^{n} \\ 1 & (-1)^{n}\end{array}\right]\) (c) \(\left[\begin{array}{ll}1+3 n & 1-4 n \\ 1+n & 1-n\end{array}\right]\) (d) \(\left[\begin{array}{ll}1+2 n & -4 n \\ 1+n & 1-2 n\end{array}\right]\)

The inverse of the matrix \(\left[\begin{array}{rrr}-0.5 & 0 & 0 \\ 0 & 4 & 0 \\\ 0 & 0 & 1\end{array}\right]\) in (a) \(\left[\begin{array}{rrr}0.5 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 1\end{array}\right]\) (b) \(\left[\begin{array}{rrr}0.5 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 1\end{array}\right]\) (c) \(\left[\begin{array}{lll}-2 & 0 & 0 \\ 0 & 0.25 & 0 \\ 0 & 0 & 1\end{array}\right]\) (d) \(\left[\begin{array}{rrr}2 & 0 & 0 \\ 0 & -0.25 & 0 \\ 0 & 0 & -1\end{array}\right]\)

If \(A=\left[\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{array}\right]\) and \(B=\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 3 \\\ 0 & 0 & 2\end{array}\right]\), then the determinant \(A B\) has the value (a) 4 (b) 8 (c) 16 (d) 32

The system of equations \(x+2 y+z=9,2 x+y+3 z=7\) can be expressed as (a) \(\left[\begin{array}{lll}1 & 2 & 1 \\ 2 & 1 & 3\end{array}\right]=\left[\begin{array}{c}9 \\\ 7\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]\) (b) \(\left[\begin{array}{lll}1 & 2 & 1 \\ 2 & 1 & 3\end{array}\right]=\left[\begin{array}{l}9 \\\ 7\end{array}\right]=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]\) (c) \(\left[\begin{array}{lll}1 & 2 & 1 \\ 2 & 1 & 3\end{array}\right]\left[\begin{array}{l}x \\ y \\\ z\end{array}\right]=\left[\begin{array}{l}9 \\ 7\end{array}\right]\) (d) none of the above.

If \(\left[\begin{array}{ll}5 & 4 \\ 1 & 1\end{array}\right] X=\left[\begin{array}{rr}1 & -2 \\ 1 & 3\end{array}\right]\), then \(X\) equals (a) \(\left[\begin{array}{rr}-3 & -14 \\ 4 & 17\end{array}\right]\) (b) \(\left[\begin{array}{rr}1 & -2 \\ 3 & 1\end{array}\right]\) (c) \(\left[\begin{array}{rr}1 & 3 \\ -2 & 1\end{array}\right]\) (d) \(\left[\begin{array}{ll}3 & -14 \\ 4 & -17\end{array}\right]\)

If \(8 x+2 y+z=0, x+4 y+z=0,2 x+y+4 z=0\), be a system of equations, then \((a)\) it is inconsistent (b) it has only the trivial solution \(x=0, y=0, z=0\). (c) it can be reduced to a single equation and so a selution does not exist. (d) determinant of the matrix of coefficients is rero.

If \(A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right], B=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]\) and \(C=\left[\begin{array}{rr}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]\), then \((a) C=A \cos \theta-B \sin \theta\) (b) \(C=A \cdot \sin \theta+B \cos \theta\) (c) \(C=A \sin \theta-B \cos \theta\) (d) \(C=A \cos \theta+B \sin \theta\).

Let \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ \alpha & 1 & 0 \\ \beta & \gamma & 1\end{array}\right]\) and \(B=\left[\begin{array}{lll}1 & 0 & 0 \\ 2 & 1 & 0 \\\ 3 & 4 & 1\end{array}\right]\), then (a) \(A\) is row equivalent to \(B\) only when \(\alpha=2, \beta=3\), and \(\gamma=4\) (b) A is mow equivalent to \(B\) only when \(\alpha \neq 0, \beta \neq 0\), and \(\gamma=0\) (c) \(\mathrm{A}\) is not row equivalent to \(B\) (d) \(A\) is row equivalent to \(B\) for all value of \(\alpha, \beta, \gamma\).

If \(A\left[\begin{array}{rr}0 & 1 \\ 2 & -1\end{array}\right]=\left[\begin{array}{rr}2 & 1 \\ -1 & 0\end{array}\right]\) where \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\), then \(A\) is (a) \(\left[\begin{array}{ll}2 & 1 \\ 0 & 0\end{array}\right]\) (b) \(\left[\begin{array}{rr}0 & 1 \\ 2 & -1\end{array}\right]\) (c) \(\left[\begin{array}{rr}2 & 1 \\ -1 & 0\end{array}\right]\) (d) \(\left[\begin{array}{cc}2 & 1 \\ -1 / 2 & -1 / 2\end{array}\right]\)

If every minor of order \(r\) of a matrix \(A\) is zero, then rank of \(A\) is (a) greater than \(r\) (b) equal to \(r\) (c) less than or equal to \(r\) (d) less than \(r\).